Dynamic analysis and optimal control of a stochastic investor sentiment contagion model considering sentiments isolation with random parametric perturbations

Investor sentiment contagion has a profound influence on economic and social development. This paper explores the diverse influences of various investor sentiments in modern society on the economy and society. It also investigates the interference of various uncertain factors on investor sentiments in the modern economy and society. On this basis, the dual-system stochastic SPA2G2R model was constructed, incorporating positive and negative sentiments, as well as a supervision and isolation mechanism. The global existence of positive solutions was established, and sufficient conditions for the disappearance and steady distribution of investor sentiment were calculated. An optimal control strategy for the stochastic model was put forward, with numerical simulation supporting the theoretical analysis results. A comparison with parameter changes in the deterministic model was also conducted. The research reveals a competitive relationship between different investor sentiments. Enhancing societal guidance mechanisms promotes positive investor sentiment contagion. Timely control by the supervisory department effectively curbs the spread of investor sentiment. Additionally, white noise promotes investor sentiment contagion, suggesting effective regulation through control of noise intensity and disturbance parameters.

• The number of individuals in the social system generally changes with time.Therefore, this paper defines B as the number of people who enter the social system.µ is defined as the rate of individuals moving out of the social system due to force majeure; • As positive and negative investor sentiments begin to disseminate in the social system, susceptible individu- als will have a probability of coming into contact with disseminators of investor sentiments.Therefore, the rate of contact with disseminators of positive investor sentiment is defined as α 1 , and the rate of contact with disseminators of negative investor sentiment is defined as α 2 .Simultaneously, susceptible individuals have a certain probability θ 1 of being influenced by the guidance mechanism and consequently becoming disseminators of positive investor sentiment; • When positive and negative investor sentiments are simultaneously disseminated in the social system, there exists a probability that disseminators of these two sentiments come into contact with each other.Therefore, this mutual contact rate of disseminators of the two investor sentiments is defined as β .Similarly, dissemina- tors of negative investor sentiment have a probability θ 2 of being influenced by guidance mechanisms, such as self-learning or publicity, and thus become disseminators of positive investor sentiment; • When the social system deems it unnecessary for the two types of investor sentiments, some disseminators of investor sentiment have certain probabilities γ 1 and γ 2 to actively choose to cease investor sentiment con- tagion due to the effectiveness of information.Other disseminators of investor sentiment have probabilities 1 and 2 of undergoing regulatory isolation by the management, transforming into isolated groups G 1 and G 2 of investor sentiment.In addition, as the disseminated investor sentiments cease to spread, the isolated groups of investor sentiment experience a reduction in the enthusiasm for investor sentiment contagion.Finally, they have probabilities ǫ 1 and ǫ 2 of choosing not to disseminate investor sentiment any longer.
In addition, the uncertain factors in social systems are commonly referred to as environmental noise.It is not scientific to study the spread of investor sentiment while ignoring random environmental noise fluctuations.
Incorporating environmental noise into deterministic models is more representative of how investor sentiment contagion in real society.The random factors added to the spread models mainly include three classical approaches: (1) Introducing Gaussian white noise into deterministic parameter perturbation models 34 .(2) Random perturbation encompassing the positive endemic equilibrium of deterministic models 35 .(3) Alternating between regimes based on the probability of Markov chains 36 .Since random perturbations in the environment may affect the contact rate under guidance mechanism and the proportion of investor sentiment disseminators under regulatory quarantine, this paper uses Gaussian white noise to generate random perturbations of θ 1 , θ 2 , 1 and 2 , and the parameters of random perturbation are expressed as follows: (1) www.nature.com/scientificreports/Here, W i (i = 1, 2, 3, 4) are independent standard Brownian motions and σ 2 i > 0(i = 1, 2, 3, 4) represent the intensities of W i (i = 1, 2, 3, 4) , respectively.In this paper, W 1 , W 2 , W 3 and W 4 represent the relationship without mutual influence between θ 1 , θ 2 , 1 and 2 , respectively.
The stochastic perturbation parameters are introduced into the deterministic model to construct a stochastic SPA2G2R model driven by Gaussian white noise, and the stochastic model can be represented as:

Existence of the global and positive solution
In the rest of this paper, let (�, F , {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfy- ing the usual conditions.And while F 0 contains all P − null sets, it is increasing and right continuous 37 .It also can be denoted as: Whether the global solution is existence is the basis of analyzing the dynamic behavior of stochastic system (2).At the same time, according to the actual situation, it is required a positive value for the dynamic model of investor sentiment contagion.The stochastic system (2) can be proved global and positive by Theorem 1.

Theorem 1
The existence of a unique positive solution (S(t), + .The probability of the solution is 1 and remains in R 5 + .

Disappearance of the information
Theorem 2 and Theorem 3 give the condition for the disappearance of the investor sentiment.The condition is expressed by intensities of noises and parameters of deterministic system.In the stochastic SPA2G2R model built in this paper, (1) Theorem 2 gives the condition for the disappearance of positive investor sentiment, (2) Theorem 3 gives the condition for the disappearance of negative investor sentiment.
Theorem 2 For any given initial value (S(0), P(0 Proof Use It ô′ s formula to calculate the differentiation of P(t) in stochastic system (2), and d ln P(t) can be written as: Thus, ln P(t) can be denoted as:

Denote
� 1 (t) and � 2 (t) are continuous local martingale.The quadratic variation of � 1 (t) and � 2 (t) can be denoted as: By exponential martingale inequality 38 , it can be known that where 0 < c < 1 , k is a random integer.Using Borel-Cantelli lemma, it is easy to know that the random integer Then, it can be obtained that noting that (9)   www.nature.com/scientificreports/Substituting Eq. ( 18) into Eq.( 17), ln P(t) can be written as: t can be obtained as: By the strong law of large numbers to the Brownian motion, let k → ∞ and then t → ∞ , it can be known that lim t can be obtained as: .
Proof Use It ô′ s formula to calculate the differentiation of A(t) in stochastic system (2), and d ln A(t) can be written as: Thus, ln A(t) can be denoted as: The quadratic variation of � 3 (t) can be denoted as: Similar to Theorem 2, for all t ∈ [0, k] , one can obtain And then, it can be obtained that noting that ( 18) ln P(t) ≤ ln P(0) ln P(t) t www.nature.com/scientificreports/Substituting Eq. ( 29) into Eq.( 28), ln A(t) can be written as: t can be obtained as: By the strong law of large numbers to the Brownian motion, let k → ∞ and then t → ∞ , it can be known that Finally, let c → 0 , lim t→∞ sup ln A(t) t can be obtained as:

A sufficient condition for the stationary distribution
Theorem 4 gives the unique stationary distribution of the existence of stochastic system (2).This also means the stability in a stochastic sense.

By the investor sentiment-existence equilibrium point
where The differential L operator to 1 can be calculated as: Vol:.( 1234567890) , it is easy to get that and then, L 1 can be expressed as: where α 1 θ 1 (S − S * ) ≥ 0 and βθ 2 (A − A * ) ≥ 0.
By simple calculation, one can get due to 1 2 (x + y) 2 ≤ x 2 + y 2 , it is easy to obtain that Similarly, L 2 can be obtained that Next, the differential L operator to 3 can be calculated as: , it is easy to get that and L 3 can be obtained as: By simple calculation, one can get due to 1 2 (x + y) 2 ≤ x 2 + y 2 , it is easy to obtain that Similarly, L 4 can be obtained that Finally, the differential L operator to 5 can be calculated as: (39) (46) (48) Vol  44), ( 49), ( 50) and (51) into Eq.( 37) to get By Eq. ( 34), the ellipsoid lies entirely in R 5 + .According to 37 , it is easy to know that stochastic system (2) has a stable stationary distribution.
Remark 2 By Theorem 4, there exist so that the solution of stochastic system (2) fluctuates around E * .Moreover, the difference between deterministic system and stochastic system (2) decreases with the values of σ 1 , σ 2 , σ 3 and σ 4 decreasing.

The stochastic optimal control model
Based on the random investor sentiment contagion model established above, the paper recognizes that positive investor sentiment significantly promotes economic and social development.Conversely, when managers need to regulate investor sentiment, effective measures of regulatory isolation can be implemented.In this view, the paper introduces two control objectives aimed at facilitating the transformation of positive investor sentiment disseminators and groups under regulatory isolation.Consequently, the four constants of proportionality in the model θ 1 , θ 2 , 1 and 2 were changed into control variables θ 1 (t), θ 2 (t), 1 (t) and 2 (t).
Hence, the objective function can be proposed as: and the objective function satisfy the state system as: (51) (52) www.nature.com/scientificreports/ The initial conditions for system (56) are satisfied: where while U is the admissible control set.0 and t f are the time interval.The control strength and importance of control measures are expressed as c 1 , c 2 , c 3 and c 4 , which are the positive weight coefficients.
Theorem 5 There exists an optimal control pair θ * 1 , θ * 2 , * 1 , * 2 ∈ U , so that the function is established as: The following five conditions must be satisfied and then the optimal control pair is existence.
(i) The set of control variables and state variables is nonempty.
(ii) The control set U is convex and closed.
(iii) The right-hand side of the state system is bounded by a linear function in the state and control variables.
(iv) The integrand of the objective functional is convex on U.
(v) There exist constants d 1 , d 2 > 0 and ρ > 1 such that the integrand of the objective functional satisfied: It is clearly that conditions (i)-(iii) established.Then, the condition (iv) can be easily established such that Next, for any t ≥ 0 , there is a positive constant M which is satisfied Hence, the optimal control can be realized.
Theorem 6 There exist adjoint variables δ 1 , δ 2 , δ 3 , δ 4 , δ 5 for the optimal control pair θ * 1 , θ * 2 , * 1 , * 2 that satisfy: With boundary conditions: In addition, the optimal control pair θ * 1 , θ * 2 , * 1 , * 2 of state system (56) can be given by: (57) S(0) = S 0 , P(0 www.nature.com/scientificreports/Proof In order to obtain the expression of optimal control system and optimal control pair, define a Hamiltonian function, which can be written as: According to the Pontyragin maximum principle, the adjoint system can be written as: and the boundary conditions of adjoint system are Then, the optimal control pair θ * 1 , θ * 2 , * 1 , * 2 can be calculated as: Remark 3 So far, the optimal control system can be got includes state system (56) with the initial conditions S(0), P(0), A(0), G 1 (0), G 2 (0) and the adjoint system (64) with boundary conditions with the optimization conditions.The optimal control system can be written as: (66) (67)   sufficient conditions for information disappearance and stable information distribution, and presents an optimal control strategy for the stochastic model.Numerical simulations were conducted to verify the probability density distribution of the stochastic model and the influence of white noise disturbance on information transmission.Furthermore, the tendencies of information transmission under various disturbance strengths were compared.
The study yields the following results: (1) White noise disturbance has the potential to promote positive investor sentiment contagion and restrain negative investor sentiment contagion.(2) As the disturbance strength increases, the randomness of the model gradually intensifies, and the fluctuation of information transmission tendency becomes more pronounced.(3) The effective control of investor sentiment contagion can be achieved by manipulating random parameters.Notably, the optimal control strategy proposed in this study differs from previous approaches, providing the optimal value calculated based on control variables.
The approach of building a non-deterministic model of investor sentiment contagion by incorporating uncertain factors into the deterministic model aligns more closely with the complexity of the real social system.This study, based on the relevant research, uses the mean field differential equation to describe the dynamic process of investor sentiment contagion.At the same time, by introducing the random factors in the social system into the deterministic model, it can better reflect the real phenomenon of the social system.In addition, the control strategy given in this paper is based on the optimal solution calculated by the optimal control model.The research findings indicate that leveraging the randomness and complexity inherent in the economy and society can greatly promote positive investor sentiment contagion, contributing to economic and social development.For investor sentiment that is deemed unnecessary, the study recommends harnessing social fluctuations and implementing timely regulatory isolation measures.
Different from previous studies, the highlights of this article are (1) In terms of research perspective, this article used the mean field differential equation model to describe the contagion mechanism of investor sentiment, Figure 7.The densities of (A) S(t), (B) P(t), (C) A(t), (D) G 1 (t) , (E) G 2 (t) change over time when σ i (i = 1, 2, 3, 4) = 0.0001 under constant control measure and optimal control.which can describe the contagion trend of investor sentiment from a microscopic perspective.(2) In terms of research methods, this article used white noise perturbation to characterize the random phenomena of social systems, and adds random parameter perturbation terms to the deterministic investor sentiment contagion model.This making the model constructed in this article more practical.(3) In terms of research results, the optimal control strategy proposed in this study differs from previous approaches, providing the optimal value calculated based on control variables.The research results of this article are different from past studies, as multiple investor sentiment exhibit a mutually inhibitory relationship during the contagion process.In addition, the control method proposed in this article can effectively promote the contagion of different investor sentiment by adjusting the random disturbance term.At the same time, the isolation of investor sentiment can quickly eliminate the contagion of various investor sentiment.
In this paper, the white noise perturbation has been used to characterize the impact of random factors in social systems on the investor sentiment contagion.And a stochastic SPA2G2R model considering different investor sentiment contagion and regulatory isolation has been constructed.White noise can clearly characterize the continuous random perturbation to the system disturbance.However, in the real social systems, the noncontinuous random perturbations are also relatively common phenomena.This paper mainly focused on the impact of continuous random perturbations on the contagion of investor sentiment, without considering the impact of non-continuous random perturbations on the contagion of investor sentiment.In future research, the non-continuous random perturbation phenomena existing in social systems will be considered.And construct an investor sentiment contagion model with non-continuous random perturbations.At the same time, the L é vy jump will be used to characterize the impact of non-continuous random perturbations on the contagion of investor sentiment.On this basis, the contagion trends of continuous and non-continuous random perturbations

Figure 9 .
Figure 9.The densities of (A) S(t), (B) P(t), (C) A(t), (D) G 1 (t) , (E) G 2 (t) change over time with different intensity of perturbation under constant control measure and optimal control.